Gradient Ricci–Yamabe Soliton on Twisted Product Manifolds

Kim, Byung Hak; Choi, Jin Hyuk; Lee, Sang Deok; Han, Chang Yong

doi:https://doi.org/10.1155/2022/5429070

Journal of Mathematics

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Research Article | Open Access

Volume 2022 | Article ID 5429070 | https://doi.org/10.1155/2022/5429070

Gradient Ricci–Yamabe Soliton on Twisted Product Manifolds

Byung Hak Kim,¹Jin Hyuk Choi,²Sang Deok Lee,³and Chang Yong Han¹

Academic Editor: R. U. Gobithaasan

Received14 Dec 2021

Revised18 May 2022

Accepted25 May 2022

Published30 Jun 2022

Abstract

In this paper, we study the twisted product manifolds with gradient Ricci–Yamabe solitons. Then, we classify and characterize the warped product and twisted product spaces with gradient Ricci–Yamabe solitons. We also study the construction of the model space of gradient Ricci–Yamabe solitons in the Riemannian product manifolds and the warped product manifolds. Finally, we study the geometric characterization of the conformally flat twisted product manifolds with gradient Ricci–Yamabe solitons.

1. Introduction

The concept of a Ricci flow was introduced by Hamilton [1], which is both generalization of Einstein metric and a special solution of the Ricci flow. Since G. Perelman solved the Poincare conjecture using Ricci flow, the study of many geometric flows and solitons has been actively conducted and have increased considerably to obtain a complete geometric classification of the manifolds and construction of model spaces.

A Riemannian metric on a complete Riemannian manifold is called a Ricci soliton if there exists a smooth vector field such that the Ricci tensor satisfies the following equation:for some constant , where is the Lie derivative with respect to [2–4]. It is said that or is a Ricci soliton if the metric on is a Ricci soliton. The Ricci soliton is called shrinking if , steady if , and expanding if . The metric of a Ricci soliton is useful in not only physics but also mathematics and is often referred to as quasi-Einstein metric [5]. If for some function on , then is called a gradient Ricci soliton with a potential function or [6]. In this case, equation (1) can be rewritten aswhere is Hessian of .

If is a function, then is called an almost gradient Ricci soliton with .

A Riemannian metric on a Riemannian manifold is called a Yamabe soliton [7–10] if there exist a smooth vector field and a constant such thatwhere is the scalar curvature of . The gradient Yamabe soliton is called shrinking if , steady if , and expanding if . When for some function on , we say that is a gradient Yamabe soliton with a potential function or . In this case, equation (3) becomes

In (4), if is a function on , then is called an almost gradient Yamabe soliton with [7, 8, 10]. When the function is constant, we say that the corresponding Yamabe soliton is a trivial Yamabe soliton. It is well known that every compact Yamabe soliton is of constant curvature and hence trivial because turns out to be constant in this case [11]. In 2019, Guler and Crasmareanu [12] introduced a scalar combination of the Ricci flow and the Yamabe flow under the name of Ricci–Yamabe map. A Riemannian metric on a Riemannian manifold with dim is said to admit a Ricci–Yamabe soliton if there exist a smooth vector field on and constants such thatwhere is the Ricci curvature [13, 14]. In particular, if for some smooth function on , we say that is a gradient Ricci–Yamabe soliton with . In this case, equation (5) becomes

The Ricci–Yamabe soliton (or gradient Ricci–Yamabe soliton) is said to be a(a)Ricci soliton (or gradient Ricci soliton) if [4, 12].(b)Yamabe soliton (or gradient Yamabe soliton) if [7, 9, 10].(c)Einstein soliton (or gradient Einstein soliton) if [3].(d)-Einstein soliton (or gradient -Einstein soliton) if [14].

The Ricci–Yamabe soliton or gradient Ricci–Yamabe soliton is said to proper if [13, 14] .

This paper is organized as follows. In Section 2, we introduce the results of geometric characterization of gradient Ricci–Yamabe soliton in the product manifold. In Section 3, we study the geometric characterization of the warped product manifold with gradient Ricci–Yamabe soliton. In Sections 4 and 5, we introduce the geometric results obtained under more generalized assumptions than the assumptions used in previous studies of the warped product manifold and the twisted product manifold , respectively.

2. Gradient Ricci–Yamabe Soliton in the Product Manifold

Let be the product manifold of the -dimensional Riemannian manifold and -dimensional Riemannian manifold . If is a generalized gradient Ricci–Yamabe soliton , thenwhere and are Levi-Civita connections for and , respectively, and are Ricci curvatures on and , respectively, and is a constant. The ranges of the indices and are and , respectively.

From the second equation of (7), we see that the potential function is repressed by for some functions on and on , respectively. Using this fact and the first and third equations of (7), we get . Since and depend only and , respectively, the quantity becomes constant, so we can put for a constant . From these facts and (7), the first and third equations reduced to and , that is, and become gradient Ricci soliton with and as potential functions, respectively.

Thus, we have the following.

Theorem 1. Let be a gradient Ricci–Yamabe soliton with . Then, we have(1)for some functionsandonand, respectively.(2) is constant.(3) is gradient Ricci soliton with a potential function .(4) is gradient Ricci soliton with a potential function .

Conversely, if and are the gradient Ricci solitons with potential functions and , respectively, then we obtainfor constants and .

If we take a function on by for a constant and let , then we see that

From (8), we get and . So,where the dimension of and . From (10), we see that and is constant if and only if is constant. Hence, equation (9) is reduced to .

Thus, we have the following.

Theorem 2. Letandbe the gradient Ricci solitons withand, respectively. Assume thatandare constants; then, the product spacebecomes a gradient Ricci–Yamabe soliton with.

If we apply Theorem 2, we can construct the model spaces of gradient Ricci–Yamabe solitons on the Riemannian product manifold.

3. Gradient Ricci–Yamabe Soliton in the Warped Product Manifold

Let the warped product manifold be a gradient Ricci–Yamabe soliton with and be a warping function depending only on . Then, we getwhere and for the coordinate in .

From (11), we have the following.

Theorem 3. If the warped product manifoldis a proper gradient Ricci–Yamabe soliton withand, then we have the following:(a)Iffor all, then we see that becomes Einstein manifold. becomes an almost gradient Ricci soliton with a potential function and a function .(b)If, then eitheris the Riemann product ofand steady gradient Ricci soliton oris Einstein.

Proof. Assume that for all , that is, depends only on . Then, we see that becomes a function on from the first equation of (11), so that and become a constant. On the other hand, from the third equation of (11), we obtainwhere . Then, equation (12) is reduced to the form for , that is, is an Einstein when and because does not guarantee that is constant.
Since depends only on , we get an equation , where we have put . Therefore, becomes an almost gradient Ricci soliton with a potential function .
If , then we see that becomes constant from the first and fourth equation of (11). From the second equation of (11), we get and that or because is a function only on . If we consider the case , then we see that becomes constant from the fourth equation of (11), that is, becomes a trivial Yamabe soliton. Moreover, we see that from the first equation of (11), so from the third equation of (11). Hence, we can state that and become a steady gradient Ricci soliton, that is, is the Riemannian product of and steady gradient Ricci soliton. Moreover, itself becomes a steady gradient Ricci soliton. For the second case , becomes constant. Then, we get from the first and third equations of (11). Hence, becomes Einstein.

Theorem 4. Let the warped product manifoldbe a proper gradient Ricci–Yamabe soliton withandfor some functionsandinand, respectively. If, then M is one of the following manifolds:(a)M is the Riemann product ofand a gradient Ricci soliton manifold and thatandbecome constant.(b)F becomes an Einstein manifold.(c)M is the Riemann product ofand an Einstein manifold.

Proof. Since , we get from the second equation of (11). From this fact, we can consider the following three cases:(i)is constant.(ii)is constant, say, and that.(iii)is constant and is constant.In case (i), becomes a Riemann product and because is constant. Moreover, we obtainfrom the first, third, and fourth equations of (11), respectively. Moreover, from the first equation of (13), we see that depends only on and that depend only on , that is, is constant and that becomes constant. Hence, the second equation of (13) is reduced to . This means that is a gradient Ricci soliton with so that we can get result (i). For case (ii), we see that . Hence, we see that the fibre becomes Einstein from the first case of Theorem 3. Finally, if we combine cases (i) and (ii), then result (iii) can be obtained.
Let us consider the case that the warped product manifold is a proper gradient Ricci–Yamabe soliton with and . Then, we getwhere . From equation (14), we see that is constant and , where . Hence, we see that is Einstein. Moreover, we getThe general solution of (15) is for some constants and . Thus, we have the following.

Theorem 5. If the warped product manifoldis a gradient Ricci–Yamabe soliton withandfor all, then the potential functionbecomesand the fibrebecomes Einstein if.

If we consider the converse of Theorem 5 and the definition of the gradient Ricci–Yamabe soliton in (6), then we obtain Theorem 6, and we can construct gradient Ricci–Yamabe soliton on the warped product manifold of and an Einstein manifold with warping function if we use Theorem 6.

Theorem 6. Letbe an Einstein manifold withandfor an arbitrary constant. Then, the warped product manifoldbecomes a gradient Ricci–Yamabe soliton with.

Proof. Since is an Einstein manifold, where is constant. It is sufficient to show the following for proof: , which is equivalent toHence, if we take , for arbitrary constant , then we can show relation (16).

Theorem 7. If the warped product manifoldis a gradient Ricci–Yamabe soliton with, then we have the following:(a)Iffor all, then we see that M becomes an Einstein. M becomes an almost gradient Ricci soliton with a potential functionand a function.(b)If, then eitheris the Riemann product ofand steady gradient Ricci soliton oris Einstein.

Proof. Now consider the case of the warped product manifold of and with warping function being a gradient Ricci–Yamabe soliton with , that is, ; then, we havewhere and
If for all , then and becomes constant from the first and third equations of (17). Hence, becomes a function on . Therefore, becomes an Einstein when and , and that becomes an almost gradient Ricci soliton with a potential function and a function .
If for all , then we get from the second equation of (17), that is, is constant or is constant. By the analogous way of the proof of Theorem 3 (b), we see that either is the Riemann product of and or is Einstein.

4. Gradient Ricci–Yamabe Soliton in the Warped Product Manifold

In this section, we consider the warped product manifold of Riemannian manifolds and . If is a gradient Ricci–Yamabe soliton with , then we havewhere and

From (18), we have the following.

Theorem 8. If the warped product spaceis a gradient Ricci–Yamabe soliton with, then we have the following:(a)Iffor all, thenis either a Riemann product ofandorbecomes Einstein.(b)Iffor all, thenis constant andbecomes Einstein.

Proof. Since for all , we see that from the second equation of (18). The relation means that is constant or is constant. In the case of being a constant, we can reduce the relation from the third equation of (18) and depends only on the base space because the quantity depends only on with the help of the first equation of (18). Hence, is constant on and becomes Einstein.
(b) If , then we see that depends only on the base space from the first equation of (18). Moreover, we obtain , and is reduced, and hence is constant. From this fact and the third equation of (18), we see that becomes an Einstein manifold.

Theorem 9. Let the warped product spacebe a gradient Ricci–Yamabe soliton withandfor some functionsandinand, respectively, thenis one of the following manifolds:(a)is the Riemann product of the almost Ricci soliton and Ricci soliton.(b)becomes an Einstein manifold.(c)is the Riemann product of the almost Ricci soliton and Einstein manifold.

Proof. If the potential function for some functions and on and , respectively, in (18), then we get from the second equation of (18). This means that is constant or is constant. If is constant, then we getfrom which we see that depends only on from the first equation of (19) and that the base space becomes almost Ricci soliton. From the second equation of (19), we see that depends only on , and hence becomes Ricci soliton. Therefore, in this case, becomes the Riemann product of the almost Ricci soliton and Ricci soliton.
If is constant, then we obtainand hence we see that is constant from the second equation of (20). If we substitute the second equation of (20) into the first equation of (20), then we get , that is, becomes Einstein.

5. Gradient Ricci–Yamabe Soliton in the Twisted Product Manifold

The twisted product of -dimension Riemannian manifold and -dimension is the product manifold with , where . In [15], the authors proved the following.

Theorem 10. If the twisted product manifoldof the Riemannian manifoldsandis conformally flat and,, thenis the warped product spaceofwith and, whereiswith the metric.

In the process of proving Theorem 10, the authors knew that the warping function can be expressed as a product of two functions and in and , respectively, that is, . Now, let us study the geometric characterization of the conformally flat twisted product manifold with gradient Ricci–Yamabe soliton. If the twisted product manifold is a gradient Ricci–Yamabe soliton with , thenwhere and

If the potential function for some functions and on and , respectively, and is conformally flat, then the warping function is expressed by according to the proof process of Theorem 10 and that the first equation of (21) becomes .

Hence, we see that the quantity depends only on . Moreover, we get from the second equation of (21), that is, is constant or is constant. Considering that is a constant as the first case, then the warping function depends only on . Hence, the Riemannian metric becomes a product metric and the first equation of (21) is reduced to and , that is, becomes an almost gradient Ricci soliton with . Considering that the second case, is a constant, we see that depends only on . Thus, we have the following.

Theorem 11. Let the twisted product manifoldbe a gradient Ricci–Yamabe soliton withand conformally flat. Iffor some functionsandonand, respectively, then the quantitydepends only onandis either Riemann product of the almost gradient Ricci soliton andor the potential functiondepends only on.

It can be seen from the structure equation of the twisted product manifold that it is established even under the condition for vector fields and on and , respectively. Thus, the condition of conformally flat in Theorem 11 can be replaced by condition .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The fourth author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIP) (NRF-2017R1E1A1A03070999).

References

R. S. Hamilton, “The ricci flow on surfaces, mathematics and general relativity (santa Cruz CA, 1986),” American Mathematical Society, vol. 71, pp. 237–262, 1988.
View at: Google Scholar
H. D. Cao, Geometry of Ricci solitons, Lecture Note, Lehigh University, Bethlehem, PA, USA, 2008.
G. Catino and L. Mazzieri, “Gradient einstein solitons,” Nonlinear Analysis, vol. 132, pp. 66–94, 2016.
View at: Publisher Site | Google Scholar
T. Ivey, “New examples of complete ricci solitons,” Proceedings of the American Mathematical Society, vol. 122, no. 1, pp. 241–245, 1994.
View at: Publisher Site | Google Scholar
U. C. De and G. C. Ghosh, “On quasi einstein manifolds,” Periodica Mathematica Hungarica, vol. 48, pp. 223–231, 2004.
View at: Publisher Site | Google Scholar
M. F. Lopez and E. G. Rio, “A note on locally conformally flat gradient ricci solitons,” Geometriae Dedicata, vol. 168, no. 1, pp. 1–7, 2014.
View at: Publisher Site | Google Scholar
J. H. Choi, B. H. Kim, and S. D. Lee, “On a classification of warped product manifolds with gradient yamabe solitons,” Communications of the Korean Mathematical Society, vol. 35, pp. 261–268, 2020.
View at: Google Scholar
B. H. Kim, J. H. Choi, and S. D. Lee, “Study on twisted product almost gradient yamabe solitons,” Journal of Mathematics, vol. 2021, Article ID 9958573, 7 pages, 2021.
View at: Publisher Site | Google Scholar
T. Seco and S. Maeta, “Classification of almost yamabe solitons in euclidean spaces,” Journal of Geometry and Physics, vol. 136, pp. 97–103, 2019.
View at: Google Scholar
W. Tokura, L. Adriano, R. Pina, and M. Barboza, “On warped product gradient yamabe solitons,” Journal of Mathematical Analysis and Applications, vol. 473, no. 1, pp. 201–214, 2019.
View at: Publisher Site | Google Scholar
P. Daskalopoulos and N. Sesum, “The classification of locally conformally flat yamabe solitons,” Advances in Mathematics, vol. 240, pp. 346–369, 2013.
View at: Publisher Site | Google Scholar
S. Guler and M. Crasmareanu, “Ricci-yamabe maps for riemannian flows and their volume variation and volume entropy,” Turkish Journal of Mathematics, vol. 43, pp. 2361–2641, 2019.
View at: Publisher Site | Google Scholar
D. Dey and P. Majhi, “Sasakian 3-metric as a generalized ricci-yamabe soliton,” Queestions Mathematica, vol. 45, no. 3, pp. 409–421, 2021.
View at: Google Scholar
X. Yi and A. Zhu, “The curvature estimate of gradient ρ einstein soliton,” Journal of Geometry and Physics, vol. 162, Article ID 104063, 2021.
View at: Publisher Site | Google Scholar
Y. Agaoka and B. H. Kim, “On conformally flat twisted product manifolds,” Memoirs of the Faculty of Liberal Arts and Education, Hiroshima University, vol. 23, pp. 1–7, 1997.
View at: Google Scholar

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Copyright © 2022 Byung Hak Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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